If = _ x 0 x 2^x-x 1- x and = _ x 0 x 2^x-x 1+x^2 - 1-x^2 , then

Step-by-step solution

aligned & = _ x 0 x 2^x-x 1- x & = _ x 0 x (2^x-1 ) 1- x = _ x 0 x (2^x-1 ) 2 ^2 x 2 & = _ x 0 1 2 2^x-1 ^2 x 2 & = 1 2 _ x 0 2^x-1 x ^2 x 2 x^2 [ divided by x] & = 1 2 _ x 0 2^x-1 ( ^2 1 2 x )^2 aligned aligned & = 1 2 [ _ x 0 2^x-1 x 1 _ x 0 ( 1 2 x x 2 )^2 1 4 ] & [ _ x 0 x x =1 ] & [ _ x 0 a^x-1 x = a ] & = 1 2 2 1 1 1 4 = 1 2 2 4 & aligned aligned & Given, = _ x 0 x 2^x-x 1+x^2 - 1-x^2 & = _ x 0 x (2^x-1 ) 1+x^2 - 1-x^2 aligned Rationalise the denominator aligned & = _ x 0 x (2^x-1 ) 1+x^2 - 1-x^2 1+x^2 + 1-x^